1/24/2005 CHEM 408 - Sp05L3 - 1MultiMulti--Electron AtomsElectron Atoms)1(ˆ21ˆ2hrZH =−∇−=1e atom:r12r1r2e1e2+Z2e atom:1222212112121ˆrrZrZH +⎟⎟⎠⎞⎜⎜⎝⎛−∇−+⎟⎟⎠⎞⎜⎜⎝⎛−∇−=||2112rrrrr−=)1(ˆh)2(ˆh)2,1(eeV)1(),,(,,Ψ=Ψϕθrlmlnall coords.of e #1)2,1(),(21Ψ=Ψ rrrreven for only 2 e, exact solutions are impossible due to Veeterm1/24/2005 CHEM 408 - Sp05L3 - 2...)2(ˆ)1(ˆ,...)2,1(ˆ++= hhH• exact & simple solutions in absence of Veebecause)...2()1(,...)2,1(jiijφφ=Ψifthen where,...)2,1(,...)2,1(,...)2,1(ˆ......... ijijijEH Ψ=Ψand• neglecting Vee, the approximate Ψ of a 2e atom is simply:),,(),,()2,1(222,,111,,222111ϕθφϕθφrrmlnmln≅Ψ222111,,,, mlnmlnEεε+≅• for example, the ground state of He would be described by:)2exp()2exp()()()2,1(2120,0,110,0,1rrNrr−−=≅Ψrrφφnormalization factor= “1s(1)1s(2)”and)()()(ˆ)(αφεαφααjjjh =... ,2 ,1=α...)2()1(...++=jiijEεε1/24/2005 CHEM 408 - Sp05L3 - 3• neglecting Veefind Eapp= -4, versus Eexp= -2.9 (+38% error)•this Eappis exact energy of Ψ of ; a better estimate uses eeappVHH −=ˆˆτdHHEappappappΨΨ=>≅<∫∗ˆˆexact H• for it is easy to show iiijiapphφεφφφ==Ψˆ with )2()1()2,1(><++=><eejiappVHεεˆ21)2()1()2,1()2()1(ττφφφφddVVjieejiappee∫∫∗∗=><• for the He atom with)2exp()2exp(21rrNapp−−=Ψ||21||4||4222111221rreedzdydxdzdydxNVrrappeerrrr−=><−−∞∞−∞∞−∞∞−∞∞−∞∞−∞∞−∫∫∫∫∫∫1/24/2005 CHEM 408 - Sp05L3 - 4• the integral is messy but doable; the result is <Veff> = 5/4 Eh.• the energy estimated by this method is E ≅ −2.75 Eh, much closer to the experimental value (-2.9 Eh).•<Ĥ>appis guaranteed to be greater than the true energy by 1ˆˆEddHHappappappappapp≥ΨΨΨΨ≡><∫∫∗∗ττThe Variational Theorem:where Ĥ is the exact Hamiltonian, E1the exact ground state E, and Ψappany (appropriate) Ψ• the variational theorem provides the basis for virtually all electronic structure calculations1/24/2005 CHEM 408 - Sp05L3 - 5• the V.T. provides a way to obtain a closer estimate of the ground state energy of He atom → vary Ψappand look for lower E• one approach is to vary size of 1e orbitals: )2exp(s1 rn−=exact He+)exp(s1 rZneff−′=′better Heapprox.• varying Zeffone finds minimum <Ĥ>appfor Zeff= 27/16, with Eapp= <Ĥ>app= -2.85 Eh(>2% err)• variational provides better description of true Ψ, which is more diffuse than He+ΨWavefunctionφ0.00.51.01.5Radial Distributionr / Å0.0 0.5 1.0 1.5 2.0 2.54πr2φ20.00.20.40.60.81.01.221.69Zeff=1.69Z=2(2)s1(1)s1′′=Ψ)2( Z=1/24/2005 CHEM 408 - Sp05L3 - 6He Atom Energies* - calculations with cc-pVTZ basis set (n=55)Energy Error Error/ Eh/ %/ kJ mol-1Vee neglected-4.0000 38% +991HF Z = 2-2.7500 -5.3% -139HF Zeff = 27/16-2.8477 -1.9% -50"HF limit"* -2.8616 -1.4% -38Hylleraas 2 param -2.8922 -0.4% -10"HF limit"* + full CI -2.9032 -0.01% -0.3Experiment -2.90341/24/2005 CHEM 408 - Sp05L3 - 7• could do better than 2% error by adding more flexibility into description of orbitals, for example: parameters nal variatio... , , , ..., , , ,with ...)exp()exp()exp()(321321332211ZZZcccrZcrZcrZcr+−+−+−=φ• but there is a limit to the accuracy of the results obtain when Ψis constrained by the orbital approximation:)()...2()1(),...,2,1(21nnnφφφ=Ψ• what’s missing is electron correlation: a dependence of the instantaneous position of one electron on the positions of the other electrons in the system• even taken to the “Hartree-Fock limit” (where additional variation no longer lowers E), this type of Ψ is still inaccurate1/24/2005 CHEM 408 - Sp05L3 - 8He Atom Energies* - calculations with cc-pVTZ basis set (n=55)Energy Error Error/ Eh/ %/ kJ mol-1Vee neglected-4.0000 38% +991HF Z = 2-2.7500 -5.3% -139HF Zeff = 27/16-2.8477 -1.9% -50"HF limit"* -2.8616 -1.4% -38Hylleraas 2 param -2.8922 -0.4% -10"HF limit"* + full CI -2.9032 -0.01% -0.3Experiment -2.90341/24/2005 CHEM 408 - Sp05L3 - 9• Hylleraas (1928) proposed the function: |)|1(),(212121rrbeNerrrrrrrr−+=Ψ−−ζζwith ζ↔Zeff and b variational parameters• minimum E for ζ = 1.85, b=0.364, with Eapp= -2.89 Eh (-.4% error)• the graphs at right show the relative prob. of finding e2 at a point along the x axis given that e1 is at the point (1,0,0).+2xye1 e2x / a-2 -1 0 1 2Relative Probability0.0010.0100.1001.000Wavefunction0.00.20.40.60.81.0e1HylleraasOrbitalApprox1/24/2005 CHEM 408 - Sp05L3 - 10He Atom Energies* - calculations with cc-pVTZ basis set (n=55)Energy Error Error/ Eh/ %/ kJ mol-1Vee neglected-4.0000 38% +991HF Z = 2-2.7500 -5.3% -139HF Zeff = 27/16-2.8477 -1.9% -50"HF limit"* -2.8616 -1.4% -38Hylleraas 2 param -2.8922 -0.4% -10"HF limit"* + full CI -2.9032 -0.01% -0.3Experiment -2.90341/24/2005 CHEM 408 - Sp05L3 - 11The Orbital Approximation• Although quantitatively inaccurate, the orbital approximation isthe conceptual basis for understanding atomic structure.• The essential idea is to think of Ψ of a multi-electron atom as the product of 1e orbitals like the wavefunctions of the 1eatom, but with orbital exponents adjusted to account for the average effects of “screening” = e-e repulsion: )()...2()1(),...,2,1()(,,)(,,)(,,22222221111nnnZmlnZmlnZmlnφφφ≅Ψ• the ground state of He in such a picture could be described as• but to go beyond He, two further ideas related to electron “spin” are required )2()1()2,1()7.1(1)7.1(1 ssφφ≅Ψ1/24/2005 CHEM 408 - Sp05L3 - 12Electron Spin & the Symmetry of Ψ1. In addition to the 3 quantum numbers resulting from the 3 spatial coordinates, electrons posses a spin coordinate which implies one further quantum number. There are only two spin states available to an electron, specified by the quantum number msupdown↑↓αβms= +1/2 ms= -1/2 2. Electronic wavefunctions (space+spin) must be antisymmetric with respect to the exchange of any pair of electrons:This requirement gives rise to the “Pauli principle”, that no two electrons in an atom can reside in the same spin-orbital, i.e. share the same {n, l, ml, ms})2,1,3()1,2,3()3,1,2(),...,3,2,1(Ψ+=Ψ−=Ψ−=Ψ n1/24/2005 CHEM 408 - Sp05L3 - 13• Incorporating e spin means that electrons reside in
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